nLab
cohomology of a category

Idea

The cohomology of a category $C$ is often defined to be the groupoid cohomology of the ∞-groupoid that is presented by the nerve of the category.

Hence for $A$ some coeffiecient ∞-groupoid – typically, but not necessarily, an Eilenberg-MacLane object $A=K(A,n)=B^{n}A$ – regarded as a Kan complex, the cohomology of $C$ in this sense is

$$H^n(C,A):={\pi }_0\mathrm{\infty }\mathrm{Grpd}(F(N(C)),A),$$H^n(C,A) := \pi_0 \infty Grpd( F(N(C)), \mathbf{A} ) \,,

where

• $N(C)$ is the nerve of $C$;

• $F(N(C))$ is the Kan fibrant replacement of $N(C)$;

• ∞Grpd is the (∞,1)-category of ∞-groupoids.

Using the standard model structure on simplicial sets this is the same as the hom-set in the homotopy category of SSet

$$\cdots ={\mathrm{Ho}}_{\mathrm{SSet}}(N(C),A).$$\cdots = Ho_{SSet}(N(C), \mathbf{A}) \,.

One can also use the Thomason model structure on Cat to say the same: due to the Quillen equivalence ${\mathrm{Cat}}_{\mathrm{Thomason}}\stackrel{\mathrm{Quillen}}{\simeq }{\mathrm{SSet}}_{\mathrm{Quillen}}$ we have for $\alpha$ any category whose groupoidification is equivalent to $A$, i.e. any cateory such that $F(N(\alpha ))\simeq A$ in ∞Grpd, we have

$$\cdots ={\mathrm{Ho}}_{{\mathrm{Cat}}_{\mathrm{Thomason}}}(C,\alpha ).$$\cdots = Ho_{Cat_{Thomason}}(C,\alpha) \,.